Problem: Solve for $x$ and $y$ by deriving an expression for $x$ from the second equation, and substituting it back into the first equation. $\begin{align*}-5x-7y &= 1 \\ 5x+9y &= 3\end{align*}$
Begin by moving the $y$ -term in the second equation to the right side of the equation. $5x = -9y+3$ Divide both sides by $5$ to isolate $x$ $x = {-\dfrac{9}{5}y + \dfrac{3}{5}}$ Substitute this expression for $x$ in the first equation. $-5({-\dfrac{9}{5}y + \dfrac{3}{5}}) - 7y = 1$ $9y - 3 - 7y = 1$ Simplify by combining terms, then solve for $y$ $2y - 3 = 1$ $2y = 4$ $y = 2$ Substitute $2$ for $y$ in the top equation. $-5x-7( 2) = 1$ $-5x-14 = 1$ $-5x = 15$ $x = -3$ The solution is $\enspace x = -3, \enspace y = 2$.